51 research outputs found
Infill topology and shape optimisation of lattice-skin structures
Lattice-skin structures composed of a thin-shell skin and a lattice infill
are widespread in nature and large-scale engineering due to their efficiency
and exceptional mechanical properties. Recent advances in additive
manufacturing, or 3D printing, make it possible to create lattice-skin
structures of almost any size with arbitrary shape and geometric complexity. We
propose a novel gradient-based approach to optimising both the shape and infill
of lattice-skin structures to improve their efficiency further. The respective
gradients are computed by fully considering the lattice-skin coupling while the
lattice topology and shape optimisation problems are solved in a sequential
manner. The shell is modelled as a Kirchhoff-Love shell and analysed using
isogeometric subdivision surfaces, whereas the lattice is modelled as a
pin-jointed truss. The lattice consists of many cells, possibly of different
sizes, with each containing a small number of struts. We propose a penalisation
approach akin to the SIMP (solid isotropic material with penalisation) method
for topology optimisation of the lattice. Furthermore, a corresponding
sensitivity filter and a lattice extraction technique are introduced to ensure
the stability of the optimisation process and to eliminate scattered struts of
small cross-sectional areas. The developed topology optimisation technique is
suitable for non-periodic, non-uniform lattices. For shape optimisation of both
the shell and the lattice, the geometry of the lattice-skin structure is
parameterised using the free-form deformation technique. The topology and shape
optimisation problems are solved in an iterative, sequential manner. The
effectiveness of the proposed approach and the influence of different
algorithmic parameters are demonstrated with several numerical examples.Comment: 20 pages, 17 figure
Adding quadric fillets to quador lattice structures
Gupta et al. [1, 2] describe a very beautiful application of algebraic geometry to lattice structures composed of quadric of revolution (quador) implicit surfaces. However, the shapes created have concave edges where the stubs meet, and such edges can be stress-raisers which can cause significant problems with, for instance, fatigue under cyclic loading. This note describes a way in which quadric fillets can be added to these models, thus relieving this problem while retaining their computational simplicity and efficiency
Isogeometric FEM-BEM coupled structural-acoustic analysis of shells using subdivision surfaces
We introduce a coupled finite and boundary element formulation for acoustic
scattering analysis over thin shell structures. A triangular Loop subdivision
surface discretisation is used for both geometry and analysis fields. The
Kirchhoff-Love shell equation is discretised with the finite element method and
the Helmholtz equation for the acoustic field with the boundary element method.
The use of the boundary element formulation allows the elegant handling of
infinite domains and precludes the need for volumetric meshing. In the present
work the subdivision control meshes for the shell displacements and the
acoustic pressures have the same resolution. The corresponding smooth
subdivision basis functions have the continuity property required for the
Kirchhoff-Love formulation and are highly efficient for the acoustic field
computations. We validate the proposed isogeometric formulation through a
closed-form solution of acoustic scattering over a thin shell sphere.
Furthermore, we demonstrate the ability of the proposed approach to handle
complex geometries with arbitrary topology that provides an integrated
isogeometric design and analysis workflow for coupled structural-acoustic
analysis of shells
Stochastic PDE representation of random fields for large-scale Gaussian process regression and statistical finite element analysis
The efficient representation of random fields on geometrically complex
domains is crucial for Bayesian modelling in engineering and machine learning.
Today's prevalent random field representations are restricted to unbounded
domains or are too restrictive in terms of possible field properties. As a
result, new techniques leveraging the historically established link between
stochastic PDEs (SPDEs) and random fields are especially appealing for
engineering applications with complex geometries which already have a finite
element discretisation for solving the physical conservation equations. Unlike
the dense covariance matrix of a random field, its inverse, the precision
matrix, is usually sparse and equal to the stiffness matrix of a Helmholtz-like
SPDE. In this paper, we use the SPDE representation to develop a scalable
framework for large-scale statistical finite element analysis (statFEM) and
Gaussian process (GP) regression on geometrically complex domains. We use the
SPDE formulation to obtain the relevant prior probability densities with a
sparse precision matrix. The properties of the priors are governed by the
parameters and possibly fractional order of the Helmholtz-like SPDE so that we
can model on bounded domains and manifolds anisotropic, non-homogeneous random
fields with arbitrary smoothness. We use for assembling the sparse precision
matrix the same finite element mesh used for solving the physical conservation
equations. The observation models for statFEM and GP regression are such that
the posterior probability densities are Gaussians with a closed-form mean and
precision. The expressions for the mean vector and the precision matrix can be
evaluated using only sparse matrix operations. We demonstrate the versatility
of the proposed framework and its convergence properties with one and
two-dimensional Poisson and thin-shell examples
An unstructured immersed finite element method for nonlinear solid mechanics.
We present an immersed finite element technique for boundary-value and interface problems from nonlinear solid mechanics. Its key features are the implicit representation of domain boundaries and interfaces, the use of Nitsche's method for the incorporation of boundary conditions, accurate numerical integration based on marching tetrahedrons and cut-element stabilisation by means of extrapolation. For discretisation structured and unstructured background meshes with Lagrange basis functions are considered. We show numerically and analytically that the introduced cut-element stabilisation technique provides an effective bound on the size of the Nitsche parameters and, in turn, leads to well-conditioned system matrices. In addition, we introduce a novel approach for representing and analysing geometries with sharp features (edges and corners) using an implicit geometry representation. This allows the computation of typical engineering parts composed of solid primitives without the need of boundary-fitted meshes.This work was partially supported by the EPSRC (second author, Grant #EP/G008531/1), by the European Research
Council (third author, Grant #ERC-2012-StG 306751), and by the Spanish Ministry of Economy and Competitiveness (third
author, Grant #DPI2015-64221-C2-1-R).This is the final version of the article. It first appeared from Springer at http://dx.doi.org/10.1186/s40323-016-0077-5
Robust topology optimisation of lattice structures with spatially correlated uncertainties
The uncertainties in material and other properties of structures are usually
spatially correlated. We introduce an efficient technique for representing and
processing spatially correlated random fields in robust topology optimisation
of lattice structures. Robust optimisation considers the statistics of the
structural response to obtain a design whose performance is less sensitive to
the specific realisation of the random field. We represent Gaussian random
fields on lattices by leveraging the established link between random fields and
stochastic partial differential equations (SPDEs). It is known that the
precision matrix, i.e. the inverse of the covariance matrix, of a random field
with Mat\'ern covariance is equal to the finite element stiffness matrix of a
possibly fractional PDE with a second-order elliptic operator. We consider the
discretisation of the PDE on the lattice to obtain a random field which, by
design, considers its geometry and connectivity. The so-obtained random field
can be interpreted as a physics-informed prior by the hypothesis that the
elliptic SPDE models the physical processes occurring during manufacturing,
like heat and mass diffusion. Although the proposed approach is general, we
demonstrate its application to lattices modelled as pin-jointed trusses with
uncertainties in member Young's moduli. We consider as a cost function the
weighted sum of the expectation and standard deviation of the structural
compliance. To compute the expectation and standard deviation and their
gradients with respect to member cross-sections we use a first-order Taylor
series approximation. The cost function and its gradient are computed using
only sparse matrix operations. We demonstrate the efficiency of the proposed
approach using several lattice examples with isotropic, anisotropic and
non-stationary random fields and up to eighty thousand random and optimisation
variables
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